There are a few approaches you can take, depending on the level of anonynimity you want. Do you just want your IP address not to show up in web server logs, or do you want it to be much more difficult to prove that you visited a website at all, even against a determined attacker? Here are some options:

Tor and Privoxy:
This combination is probably the best that is reasonably achievable, but slower than other methods. Rather than me explaining how it works, take a look at http://tor.eff.org

Anonymizing Services:
www.secure-tunnel and metropipe.net are a couple. They act as proxies that you can use SSH to connect to and tunnel your requests through them. This is less secure as you have to trust them, and a determined attacker could probably defeat this level of anonynimity pretty easily. On the plus side, they are faster than Tor, but they cost about $7/month.

CGI proxy sites. You go to those sites (such as www.trickmy.net) and simply enter the URL you wish to browse annonymously.....the site you browse will not see your IP address if you have entered thru one of these sites. Easiest way to use proxies and remain annonymous. Nothing to download. All browser based.

Yes, Tor can prevent you from being monitored by your network admin. Depending on what level of monitoring they do, they may notice that you are using Tor, but they won't be able to tell what you are using Tor for. The downside is speed. Tor is a bit slower than browsing the internet normally, as your connections go through three Tor servers, which adds latency and reduces bandwidth... If you can get more specific info on what logging/monitoring they do, perhaps we can come up with a level of security that is "just enough" to defeat the monitoring but maintain as much bandwidth and low latency as possible...

RE: CGI Proxies: These will prevent the site you visit from getting your IP (as long as they don't download the images directly from the server to your computer, as I've seen a few do in order to save the CGI site's bandwidth!). Keep in mind that most of them do not offer encryption. I don't really consider them to be anywere near as secure as Tor.

If you downloaded the bundle then it is already configured to use the application proxy and you simply have to go to:

>tools within IE
>internet options
>connections tab
>settings button
>tick the ‘use a proxy server for this…’ box
>enter the 127.0.0.1 for the address
>enter 8118 for the port

you can push the advanced button for a little more granularity but those are the basic settings.

If you downloaded them separately then you also need to add a line to privoxys configuration file- you will find said line in the tor documentation. I can’t remember it off the top of my head as I now use the bundle.

There's a posablitity that IT also has remote desktop or something like VNC installed. If that's the case then they can still see what you're doing, or other spying software which is hidden on your computer and reports what you do. That type of software is becoming more commonplace lately. Also if the network admin sees a boat load of encrypted traffic(ie. more than you just using a secure website to order stuff) they will become suspicous and might ask you about it.

Generally the rule of thumb is IT is not your enemy, they have rules and restrictions in place for a reason such as data security. I'll admit though there are the hardasses taht won't even let you use USB drives, or run anything not on port 80, but overall it's not usually unreasonable. You might find if you talk to them, whatever you're tryign to do their ok with. We're people too.

I'd recommend simply not visiting sites at work that you don't want your IT department to see. With increase big brother and corporate monitoring, they create slicker methods all the time. Suffice it to say that a close friend of mine works for a company that creates hardware-level corporate monitoring devices that are very hard to fool.